Angular Velocity

10 Key excerpts on "Angular Velocity"

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  eBook - PDF

  Halliday and Resnick's Principles of Physics

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  Angular Velocity Suppose that our rotating body is at angular position θ 1 at time t 1 and at angular position θ 2 at time t 2 as in Fig. 10-4. We define the average Angular Velocity of the body in the time interval Δt from t 1 to t 2 to be ω avg = θ 2 − θ 1 t 2 − t 1 = Δθ Δt , (10-5) where Δθ is the angular displacement during Δt (ω is the lowercase omega). An angular displacement in the counterclockwise direction is positive, and one in the clockwise direction is negative. 221 10-1 ROTATIONAL VARIABLES Figure 10-4 The reference line of the rigid body of Figs. 10-2 and 10-3 is at angular posi- tion θ 1 at time t 1 and at angular position θ 2 at a later time t 2 . The quantity Δθ (= θ 2 − θ 1 ) is the angular displacement that occurs during the interval Δt (= t 2 − t 1 ). The body itself is not shown. x y Rotation axis O θ 1 θ 2 θ At t 2 At t 1 Reference line This change in the angle of the reference line (which is part of the body) is equal to the angular displacement of the body itself during this time interval. Δ The (instantaneous) Angular Velocity ω, with which we shall be most con- cerned, is the limit of the ratio in Eq. 10-5 as Δt approaches zero. Thus, ω = lim Δt →0 Δθ Δt = dθ dt . (10-6) If we know θ(t), we can find the Angular Velocity ω by differentiation. Equations 10-5 and 10-6 hold not only for the rotating rigid body as a whole but also for every particle of that body because the particles are all locked together. The unit of Angular Velocity is commonly the radian per second (rad/s) or the revolution per second (rev/s). Another measure of Angular Velocity was used during at least the first three decades of rock: Music was produced by vinyl (phonograph) records that were played on turntables at “33 1 3 rpm” or “45 rpm,” meaning at 33 1 3 rev/min or 45 rev/min. If a particle moves in translation along an x axis, its linear velocity v is either positive or negative, depending on its direction along the axis.

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  Introductory Physics for the Life Sciences: Mechanics (Volume One)

  • David V. Guerra(Author)
  • 2023(Publication Date)
  • CRC Press

    (Publisher)

  If the rigid object was rotating in a clockwise direction as observed above, you would need to point the thumb of your right hand at the page and then you could rotate the fingers of your right hand in a clockwise direction; this rotation is in the -z-direction. So, with objects rotating in the x-y plane, the angular displacement vector will point in the –z-direction or the +z-direction.

  A total angular displacement from

  θ ⇀

  0

  at time t0 to an angular displacement of

  θ ⇀

  1

  at time t1 is given in equation (13.2) as:

  θ ⇀

  01

  =

  θ ⇀

  1

    −  

  θ ⇀

  0

  (13.2)

  The SI unit for angular displacement is the radian (rad).

  Example (Angular Displacement) What is the magnitude of the angular displacement, in radians, of a bicycle wheel that completes 20 rotations?

  Solution: θ01 = 20 rotations = (20 rotations)

  2 π   radians

  rotation

  = 125.66 radians

  13.2.3 Angular Velocity

  The Angular Velocity of an object is how fast and which way the angular position of the object is changing. Equivalently, the Angular Velocity of an object is how fast and which way that object is spinning at a particular instant in time. It is represented by a vector (

  ω ⇀

  ) with a magnitude of the spin rate of the object and whose direction is along the axis of rotation that corresponds to the sense of rotation by the right-hand rule. The Angular Velocity of an object is sometimes referred to as the instantaneous Angular Velocity of that object to distinguish it from the average Angular Velocity of the object over some time interval. The average value-with-direction of the Angular Velocity

  ω ⇀

  over some time interval, say from time t0 to time t1 , is obtained by dividing the total angular displacement

  θ ⇀

  01

  that occurs during that time interval by the duration of the time interval t01 = t1 − t0 and is presented in equation (13.3) as:

  ω ⇀

  Avg0 1

    =  

  θ ⇀

  01

  /

  t 01

  (13.3)

  The SI unit for Angular Velocity is the radian per second (rad/s). A common non-SI unit of Angular Velocity is rpm (revolutions per minute).

  Example (Angular Velocity) The crankshaft of a car engine is spinning at a rate of 2,500 rpm. What is the magnitude of the Angular Velocity of the crankshaft in radians per second?

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  Introduction to Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  The average linear velocity v B was defined as the linear dis- placement D x B of the object divided by the time Dt required for the displacement to occur, or v B 5 D x B / Dt (see Equation 2.2). We now introduce the analogous idea of Angular Velocity to describe the motion of a rigid object rotating about a fixed axis. The average Angular Velocity v (Greek letter omega) is defined as the angular displacement Du 5 u 2 u 0 divided by the elapsed time Dt during which the displacement occurs. Definition of Average Angular Velocity 5 Angular displacement Elapsed time v 5 u 2 u 0 t 2 t 0 5 Du Dt (8.2) SI Unit of Angular Velocity: radian per second (rad/s) Average Angular Velocity The SI unit for Angular Velocity is the radian per second (rad/s), although other units such as revolutions per minute (rev/min or rpm) are also used. In agreement with the sign con- vention adopted for angular displacement, Angular Velocity is positive when the rotation is counterclockwise and negative when it is clockwise. Example 3 shows how the concept of average Angular Velocity is applied to a gymnast. The instantaneous Angular Velocity v is the Angular Velocity that exists at any given instant. To measure it, we follow the same procedure used in Chapter 2 for the instanta- neous linear velocity. In this procedure, a small angular displacement Du occurs during a small time interval Dt. The time interval is so small that it approaches zero ( Dt B 0), and in this limit, the measured average Angular Velocity, v 5 Du/ Dt , becomes the instanta- neous Angular Velocity v: v 5 lim D t B0 v 5 lim D t B0 Du Dt (8.3) The magnitude of the instantaneous Angular Velocity, without reference to whether it is a positive or negative quantity, is called the instantaneous angular speed. If a rotating object has a constant Angular Velocity, the instantaneous value and the average value are the same.

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  eBook - PDF

  Fundamentals of Physics

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  10.1.11 Given a graph of Angular Velocity versus time, determine the instantaneous angular acceleration at any particular time and the average angular acceleration between any two particular times. 10.1.12 Calculate a body’s change in Angular Velocity by integrating its angular acceleration function with respect to time. 10.1.13 Calculate a body’s change in angular position by integrating its Angular Velocity function with respect to time. Key Ideas ● To describe the rotation of a rigid body about a fixed axis, called the rotation axis, we assume a reference line is fixed in the body, perpendicular to that axis and rotating with the body. We measure the angular position θ of this line relative to a fixed direction. When θ is measured in radians, θ = s _ r (radian measure), where s is the arc length of a circular path of radius r and angle θ. ● Radian measure is related to angle measure in revolutions and degrees by 1 rev = 360° = 2π rad. ● A body that rotates about a rotation axis, changing its angular position from θ 1 to θ 2 , undergoes an angular displacement Δθ = θ 2 − θ 1 , where Δθ is positive for counterclockwise rotation and negative for clockwise rotation. ● If a body rotates through an angular displacement Δθ in a time interval Δt, its average Angular Velocity ω avg is ω avg = Δθ _ Δt . The (instantaneous) Angular Velocity ω of the body is ω = dθ _ dt . C H A P T E R 1 0 271 10.1 ROTATIONAL VARIABLES What Is Physics? As we have discussed, one focus of physics is motion. However, so far we have examined only the motion of translation, in which an object moves along a straight or curved line, as in Fig. 10.1.1a. We now turn to the motion of rotation, in which an object turns about an axis, as in Fig. 10.1.1b. You see rotation in nearly every machine, you use it every time you open a beverage can with a pull tab, and you pay to experience it every time you go to an amusement park.

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  eBook - ePub

  Doing Physics with Scientific Notebook

  A Problem Solving Approach

  • Joseph Gallant(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Evaluate Numerically to find the number of times the record turns while it’s slowing down.

  The record turns about 2.8 times while it’s slowing down.

  When the angular acceleration is not constant, you must use calculus to describe the object’s rotational motion. The Angular Velocity is the rate of change of the angular displacement.

  (7.9)

  If you know the angular position as a function of time, you can calculate the angular speed.

  The angular acceleration is the rate of change of angular speed.

  (7.10)

  If you know the angular speed as a function of time, you can calculate the angular acceleration. From these definitions we can derive four basic equations which describe the angular position and speed of objects rotating with a varying acceleration.

  (7.11a)

  (7.11b)

  (7.11c)

  (7.11d)

  For varying angular acceleration, the average Angular Velocity does not equal the arithmetic average of the initial and final angular velocities.

  The Compact Disk

  The rotational motion of a CD is an excellent example of nonuniform rotational motion with a time-dependent acceleration. To read the information on a record, a needle starts at the outer edge and follows a single spiral track while the disk rotates at a constant angular speed. To read the information on a CD, a laser starts at the inner radius and follows a single spiral track while the disk rotates at a decreasing angular speed. This ensures the data-read rate, which is proportional to the linear speed, is constant. [31]

  The data on a typical CD is stored between R0 = 2.3 cm and Rf = 5.8 cm, and the spacing between tracks is Δr = 1.6 μm/rev. We can use Evaluate Numerically

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  Higher Engineering Science

  • William Bolton(Author)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  The instantaneous Angular Velocity ω is the change in angular displacement with time when the time interval tends to zero. It can be expressed as:

  [20]

  3 Angular accelerationThe average angular acceleration over some time interval is the change in Angular Velocity during that time divided by the time:

  [21]

  The unit is rad/s2 . The instantaneous angular acceleration a is the change in Angular Velocity with time when the time interval tends to zero. It can be expressed as:

  [22]

  4.4.1 Motion with constant angular acceleration

  For a body rotating with a constant angular acceleration α, when the Angular Velocity changes uniformly from ω0 to co in time t, as in Figure 4.19 , equation [21 ] gives:

  Figure 4.19 Uniformly accelerated motion

  and hence:

  ω = ω0 + at

  [23]

  The average Angular Velocity during this time is ½(ω + ω0 ) and thus if the angular displacement during the time is θ:

  Substituting for co using equation [23 ]:

  Hence:

  θ = ω0 t + ½at2

  [24]

  Squaring equation [23 ] gives:

  Hence, using equation [24 ]:

  [25]

  Example

  An object which was rotating with an Angular Velocity of 4 rad/s is uniformly accelerated at 2 rad/s. What will be the Angular Velocity after 3 s?

  Using equation [23 ]:

  ω = ω0 + at = 4 + 2 × 3 = 10 rad/s

  Example

  The blades of a fan are uniformly accelerated and increase in frequency of rotation from 500 to 700 rev/s in 3.0 s. What is the angular acceleration?

  Since ω = 2πf, equation [23 ] gives:

  2π × 700 = 2π × 500 + a × 3.0

  Hence a = 419 rad/s2 .

  Example

  A flywheel, starting from rest, is uniformly accelerated from rest and rotates through 5 revolutions in 8 s. What is the angular acceleration?

  The angular displacement in 8 s is 2π × 5 rad. Hence, using equation [24 ], i.e. θ = ω0 t + ½at2 :

  2π × 5 = 0 + ½a × 82

  Hence the angular acceleration is 0.98 rad/s2 .

  Revision

  13 A flywheel rotating at 3.5 rev/s is accelerated uniformly for 4 s until it is rotating at 9 rev/s. Determine the angular acceleration and the number of revolutions made by the flywheel in the 4 s.

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  Classical Mechanics

  • Tom W B Kibble, Frank H Berkshire(Authors)
  • 2004(Publication Date)
  • ICP

    (Publisher)

  Chapter 5

  Rotating Frames

  Hitherto, we have always used inertial frames, in which the laws of motion take on the simple form expressed in Newton’s laws. There are, however, a number of problems that can most easily be solved by using a non-inertial frame. For example, when discussing the motion of a particle near the Earth’s surface, it is often convenient to use a frame which is rigidly fixed to the Earth, and rotates with it. In this chapter, we shall find the equations of motion with respect to such a frame, and discuss some applications of them.

  5.1Angular Velocity; Rate of Change of a Vector

  Let us consider a solid body which is rotating with constant Angular Velocity ω about a fixed axis. Let n be a unit vector along the axis, whose direction is defined by the right-hand rule: it is the direction in which a right-hand-thread screw would move when turned in the direction of the rotation. Then we define the vector Angular Velocity ω to be a vector of magnitude ω in the direction of n: ω = ωn. Clearly, Angular Velocity, like angular momentum, is an axial vector (see §3.3 ).

  For example, for the Earth, ω is a vector pointing along the polar axis, towards the north pole. Its magnitude is equal to 2π divided by the length of the sidereal day (the rotation period with respect to the fixed stars, which is less than that with respect to the Sun by one part in 365), that is

  If we take the origin to lie on the axis of rotation, then the velocity of a point of the body at position r is given by the simple formula

  To prove this, we note that the point moves with Angular Velocity ω around a circle of radius ρ = r sin θ (see Fig. 5.1 ). Thus its speed is

  Fig. 5.1

  Moreover, the direction of v is that of ω ∧ r; for clearly, v is perpendicular to the plane containing ω and r

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  An Introduction to Mathematics for Engineers

  Mechanics

  • Stephen Lee(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  , and d d 2 t 2 as .. . Figure 12.1 shows a particle P moving round the circumference of a circle of radius r , centre O. At time t , the position vector OP ⎯→ of the particle makes an angle (in radians) with the fixed direction OA ⎯→ . The arc length AP is denoted by s . Figure 12.1 12.2 Angular speed Using this notation, s r . Differentiating this with respect to time using the product rule gives: d d s t r d d t d d r t . Since r is constant for a circle, d d r t 0, so the rate at which the arc length increases is: d d s t r d d t or s . r . . In this equation s . is the speed at which P is moving round the circle (often denoted by v ), and . is the rate at which the angle is increasing, i.e. the rate at which the position vector OP ⎯→ is rotating. The quantity d d t , or . , can be called the Angular Velocity or the angular speed of P. In more advanced work, Angular Velocity is treated as a vector, whose direction is taken to be that of the axis of rotation. In this chapter, d d t is often referred to as angular speed, but is given a sign: positive when is increasing (usually anticlockwise) and negative when is decreasing (usually clockwise). ➀ O P r s θ A CIRCULAR MOTION 253 Angular speed is often denoted by , the Greek letter omega. So the equation s . r . may be written as v r . Notice that for this equation to hold, must be measured in radians, so the angular speed is measured in radians per second or rad s 1 . Figure 12.2 shows a disc rotating about its centre, O, with angular speed . The line OP represents any radius. Figure 12.2 Every point on the disc describes a circular path, and all points have the same angular speed. However the actual speed of any point depends on its distance from the centre: increasing r in the equation v r increases v . You will appreciate this if you have ever been at the end of a rotating line of people in a dance or watched a body of marching soldiers wheeling round a corner.

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  Physics for Scientists and Engineers

  Foundations and Connections, Extended Version with Modern Physics

  • Debora Katz(Author)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  CONCEPT EXERCISE 12.3 Table 12.1 gives the angular position of a rotating bottle (Fig. 12.11) at five instants in time. a. Sketch a motion diagram for the bottle along with a coordinate system. It is not necessary to draw the actual bottle; instead, draw the reference line at each time corresponding to positions A through E. b. What is the angular displacement from A to E? Is the rotation clockwise or counterclockwise? Angular Velocity The average angular speed and instantaneous angular speed are analogous to average and instantaneous translational speed. Average and instantaneous angular speed are represented by the lowercase Greek letter omega, v. The average angular speed ω av over a time interval Dt is the angular displacement divided by the time interval: v av 5 u f 2 u i t f 2 t i 5 Du Dt (12.3) The instantaneous angular speed comes from taking the limit of Equation 12.3 as the time interval goes to zero: v 5 lim Dt S0 Du Dt 5 du dt (12.4) If we measure u in radians, the SI units of ω av and v are radians per second, or rad / s . The instantaneous angular speed is the magnitude of the instantaneous Angular Velocity vector. (We usually drop the word “instantaneous”) Just as the direction of v u refers to the specific coordinate system we choose, so does the direction of v u . The direction in which we measure angular positions is chosen to be consistent with direction of angular displacement. Neither angular displacement nor average angular speed is a vector. Angular Velocity ★ Major Concept s r u FIGURE 12.10 y x u z FIGURE 12.11 TABLE 12.1 Angular position of the rotating bottle at various time instants. Position t (s) u (rad) A 0 0.75 B 2 3.45 C 4 5.95 D 6 8.25 E 8 10.05 Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 336 CHAPTER 12 Rotation I: Kinematics and Dynamics All content on this page is © Cengage Learning. FIGURE 12.12 A. Aaron and Hannah watch a rotating potato.

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  Physics for Scientists and Engineers with Modern Physics

  • Raymond Serway, John Jewett(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Equation 10.10 shows that the tangential speed of a point on the rotat- ing object increases as one moves outward from the center of rotation, as we would intuitively expect. For example, the outer end of a swinging golf club moves much faster than a point near the handle. We can relate the angular acceleration of the rotating rigid object to the tangen- tial acceleration of the point P by taking the time derivative of v in Equation 10.10: a t 5 dv dt 5 r dv dt a t 5 r a (10.11) That is, the tangential component of the translational acceleration of a point on a rotating rigid object equals the point’s perpendicular distance from the axis of rotation multiplied by the angular acceleration. In Section 4.4, we found that a point moving in a circular path undergoes a radial acceleration a r directed toward the center of rotation and whose magnitude is that of the centripetal acceleration v 2 / r (Fig. 10.5). Because v 5 rv for a point P on a rotating object, we can express the centripetal acceleration at that point in terms of angular speed as we did for a particle moving in a circular path in Equation 4.25: a c 5 v 2 r 5 r v 2 (10.12) The total acceleration vector at the point is a S 5 a S t 1 a S r , where the magnitude of a S r is the centripetal acceleration a c . Because a S is a vector having a radial and a tan- gential component, the magnitude of a S at the point P on the rotating rigid object is a 5 Ïa t 2 1 a r 2 5 Ïr 2 a 2 1 r 2 v 4 5 r Ïa 2 1 v 4 (10.13) Q UICK QUIZ 10.3 Ethan and Rebecca are riding on a merry-go-round. Ethan rides on a horse at the outer rim of the circular platform, twice as far from the center of the circular platform as Rebecca, who rides on an inner horse.

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